Potential Difference

What Is Potential Difference?

Potential difference is the amount of energy transferred, or work done, per unit charge as charge moves between two points in an electric circuit or electric field.

Potential difference is commonly called:

voltage

It is represented by:

V

The basic relationship is:

V = W/Q

where:

• V = potential difference (V)
• W = work done or energy transferred (J)
• Q = charge transferred (C)

This means that voltage tells us how much energy is transferred for each coulomb of charge.

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The Volt

The SI unit of potential difference is the:

volt (V)

From:

V = W/Q

we obtain:

1 V = 1 J/C

Therefore:

1 volt means 1 joule of energy transferred per coulomb of charge.

For example, if a component has a potential difference of:

6 V

then each coulomb of charge transfers:

6 J

of energy as it passes through the component.

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Interpreting Voltage

Voltage is sometimes described informally as the electrical "push" that drives current.

This can be useful, but a more precise description is:

potential difference represents an energy difference per unit charge.

A potential difference creates an electric field within a conductor.

That electric field exerts forces on charged particles and causes a net movement of charge.

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Potential Difference and Electric Potential

Electric potential describes energy per unit charge at a particular location.

Potential difference compares the electric potentials at two locations.

For points A and B:

ΔV = V_B − V_A

Therefore, potential difference tells us how electric potential changes between two points.

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A Gravitational Analogy

Potential difference can be compared loosely with a difference in height.

Water at a higher gravitational position can move downward and transfer energy.

Similarly, charges moving through an electric potential difference can transfer energy.

However, voltage is not literally a height or pressure.

It is:

energy transferred per unit charge

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Voltage and Charge Movement

When a potential difference exists across a conductor, an electric field is established within the conductor.

This electric field exerts forces on mobile charges.

As a result:

potential difference → electric field → charge movement

If a complete conducting path exists, this can produce an electric current.

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Conventional Current

By convention, current direction is defined as the direction positive charge would move.

Therefore:

conventional current flows from higher potential toward lower potential through passive circuit components

In metallic wires, however, the mobile charge carriers are electrons.

Electrons move in the opposite direction to conventional current.

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Why Electrons Move the Opposite Way

Electrons have negative charge.

The electric field direction is defined as the direction of force on a positive test charge.

Because electrons are negatively charged, the electrical force on them acts opposite to the electric field.

Therefore:

conventional current direction ≠ electron drift direction

in a metal conductor.

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Batteries and Potential Difference

A battery maintains a potential difference between its terminals.

Chemical reactions inside the battery transfer energy to charges.

For example, a:

9 V battery

provides approximately:

9 J of energy per coulomb

of charge moved through the battery under idealized conditions.

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Energy Gain in a Battery

When charge moves through a battery from lower electric potential to higher electric potential, it gains electric potential energy.

For:

Q = 1 C

and:

V = 12 V

the energy transferred is:

W = QV

W = 1 × 12

W = 12 J

The battery has transferred 12 J of energy to the charge.

---

Energy Transfer in a Component

When charge passes through a component such as a resistor or lamp, electrical energy can be transferred into other forms.

For example:

Lamp

electrical energy → light + thermal energy

Heater

electrical energy → thermal energy

Motor

electrical energy → kinetic energy + thermal energy + sound

Speaker

electrical energy → sound + thermal energy

The potential difference across the component describes the energy transferred per coulomb.

---

Rearranging the Voltage Equation

Starting with:

V = W/Q

we can rearrange it.

For energy or work:

W = QV

For charge:

Q = W/V

These three forms are useful for solving different types of problems.

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Example: Calculating Potential Difference

Suppose:

W = 24 J

Q = 4 C

Use:

V = W/Q

V = 24/4

V = 6 V

Therefore, 6 J of energy is transferred for every coulomb of charge.

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Example: Calculating Energy

A charge of:

Q = 5 C

passes through a potential difference of:

V = 12 V

Use:

W = QV

W = 5 × 12

W = 60 J

Therefore:

60 J of energy is transferred.

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Example: Calculating Charge

A device transfers:

W = 180 J

when operating across:

V = 12 V

Use:

Q = W/V

Q = 180/12

Q = 15 C

Therefore:

15 C of charge passed through the device.

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Charge and Current

Charge transferred is related to current by:

Q = It

where:

• Q = charge (C)
• I = current (A)
• t = time (s)

This equation can be combined with:

W = QV

to give:

W = VIt

This allows energy transfer to be calculated from:

• voltage
• current
• time

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Example: Voltage, Current and Energy

A heater operates at:

V = 12 V

with current:

I = 3 A

for:

t = 20 s

First find charge:

Q = It

Q = 3 × 20

Q = 60 C

Then:

W = QV

W = 60 × 12

W = 720 J

Therefore:

720 J of electrical energy is transferred.

---

Direct Calculation Using W = VIt

The previous problem can be solved directly:

W = VIt

W = 12 × 3 × 20

W = 720 J

Both methods give the same result.

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Potential Difference and Power

Power is the rate of energy transfer:

P = W/t

Since:

W = VIt

then:

P = VIt/t

Therefore:

P = VI

This is one of the most important equations in electricity.

Voltage therefore connects directly to both:

energy transfer

and:

power

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Example: Electrical Power

A motor operates at:

V = 24 V

and draws:

I = 2.5 A

Power:

P = VI

P = 24 × 2.5

P = 60 W

The motor receives electrical energy at a rate of:

60 J/s

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Potential Difference and Resistance

For an ohmic conductor:

V = IR

where:

• V = potential difference
• I = current
• R = resistance

Therefore:

I = V/R

For a fixed resistance, increasing the potential difference increases the current.

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Example: Voltage and Current

A resistor has:

R = 20 Ω

and potential difference:

V = 10 V

Current:

I = V/R

I = 10/20

I = 0.50 A

If the potential difference doubles to:

20 V

then:

I = 20/20

I = 1.0 A

For the same resistance, doubling voltage doubles current.

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Potential Difference Across a Resistor

When charge moves through a resistor, electrical energy is transferred mainly into thermal energy.

If the potential difference is:

10 V

then every coulomb passing through the resistor transfers:

10 J

of electrical energy.

This energy increases the internal energy of the resistor and its surroundings.

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Potential Rise and Potential Drop

When moving around a circuit, we can describe changes in potential.

Through a battery

There is a:

potential rise

because energy is transferred to the charges.

Through a resistor

There is a:

potential drop

because energy is transferred from the electrical system to thermal energy.

Through a motor

There is a potential drop because electrical energy is transferred into kinetic energy and other forms.

---

Potential Difference Around a Simple Circuit

Consider a:

12 V battery

connected to one resistor.

Ideally, the resistor has:

12 V

across it.

The battery provides:

12 J/C

and the resistor transfers:

12 J/C

into other forms.

Energy is conserved.

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Voltage in a Series Circuit

In a series circuit, the supply potential difference is shared between the components.

For example:

V_supply = 12 V

V₁ = 4 V

V₂ = 8 V

Then:

V_supply = V₁ + V₂

12 = 4 + 8

This reflects conservation of energy.

Each coulomb gains 12 J from the source and then transfers:

4 J + 8 J = 12 J

through the components.

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Voltage in Parallel Circuits

Components connected in parallel have the same potential difference across them.

If three lamps are connected directly in parallel across a:

12 V battery

then each branch has approximately:

12 V

across it.

Each coulomb passing through any branch can therefore transfer:

12 J

through that branch.

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Measuring Potential Difference

Potential difference is measured using a:

voltmeter

A voltmeter is connected:

in parallel

with the component being measured.

This allows the meter to compare the electric potentials on opposite sides of the component.

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Why a Voltmeter Is Connected in Parallel

Potential difference is a comparison between two locations.

Therefore, the voltmeter must connect to:

both sides of the component

A good voltmeter has very high resistance so that it draws very little current and causes minimal disturbance to the circuit.

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Voltmeter vs Ammeter

A voltmeter measures:

potential difference

and is connected:

in parallel

An ammeter measures:

current

and is connected:

in series

Confusing these connections is a common circuit error.

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Potential Difference in an Electric Field

Potential difference also exists between locations in an electric field.

For a uniform electric field:

|ΔV| = Ed

where:

• E = electric field strength (V/m)
• d = distance along the field direction (m)

Therefore:

E = |ΔV|/d

A stronger field produces a larger change in potential over the same distance.

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Example: Uniform Electric Field

Two parallel plates are:

0.020 m

apart.

The electric field strength is:

5000 V/m

Potential difference:

|ΔV| = Ed

|ΔV| = 5000 × 0.020

|ΔV| = 100 V

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Work Done in an Electric Field

Suppose a charge:

Q = 2.0 × 10⁻⁶ C

moves through a potential difference of:

500 V

Energy transferred:

W = QV

W = (2.0 × 10⁻⁶)(500)

W = 1.0 × 10⁻³ J

or:

W = 1.0 mJ

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Example with Milliamps

A device operates at:

V = 9 V

Current:

I = 250 mA

for:

t = 60 s

Convert:

250 mA = 0.250 A

Energy:

W = VIt

W = 9 × 0.250 × 60

W = 135 J

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Example with Microcoulombs

A charge of:

250 μC

moves through:

400 V

Convert:

250 μC = 250 × 10⁻⁶ C

Then:

W = QV

W = (250 × 10⁻⁶)(400)

W = 0.100 J

Unit conversion is especially important in electrical calculations.

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Choosing the Correct Equation

Use:

V = W/Q

when given energy and charge.

Use:

W = QV

when calculating energy transfer.

Use:

Q = W/V

when finding charge.

Use:

Q = It

when current and time are given.

Use:

W = VIt

when voltage, current, and time are known.

Use:

P = VI

when calculating electrical power.

Use:

V = IR

for voltage, current, and resistance in an ohmic component.

Use:

|ΔV| = Ed

for a uniform electric field.

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Multi-Step Example

A 6 V motor draws:

2 A

for:

5 minutes.

Calculate the electrical energy transferred.

First convert time:

5 min = 300 s

Use:

W = VIt

W = 6 × 2 × 300

W = 3600 J

or:

W = 3.6 kJ

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Alternative Solution

First calculate charge:

Q = It

Q = 2 × 300

Q = 600 C

Then:

W = QV

W = 600 × 6

W = 3600 J

Again:

W = 3.6 kJ

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Comparing Two Devices

Device A operates across:

6 V

Device B operates across:

12 V

Each coulomb passing through Device A transfers:

6 J

Each coulomb passing through Device B transfers:

12 J

Therefore, Device B transfers twice as much energy per coulomb.

This does not automatically mean Device B uses twice as much total energy because total energy also depends on the amount of charge transferred.

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Voltage Does Not Equal Energy

Voltage is not measured in joules.

Voltage is:

energy per unit charge

For example:

12 V = 12 J/C

To calculate actual energy transferred, we also need to know how much charge moves:

W = QV

This distinction is essential.

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Voltage Does Not Equal Current

Voltage and current are different quantities.

Voltage

Measures:

energy transferred per unit charge

Unit:

volt (V)

Current

Measures:

rate of flow of charge

I = Q/t

Unit:

ampere (A)

Voltage can cause current to flow when there is a suitable conducting path, but voltage and current are not the same thing.

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What Determines Current?

The current depends on both:

• potential difference
• resistance

For an ohmic conductor:

I = V/R

Therefore:

larger V → larger current if R is constant

larger R → smaller current if V is constant

A large voltage does not automatically mean a large current if resistance is also very large.

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Potential Difference and Energy Conservation

Potential difference provides an important way to understand energy conservation in circuits.

Consider a circuit with:

12 V source

and two components having drops of:

5 V and 7 V

Each coulomb gains:

12 J

from the source.

Then transfers:

5 J + 7 J = 12 J

through the components.

Therefore:

energy gained = energy transferred

This is the energy basis of Kirchhoff's Voltage Law.

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Kirchhoff's Voltage Law

Around a complete closed loop:

total potential rises = total potential drops

or equivalently:

ΣV = 0

This follows from conservation of energy.

A charge returning to its original point in a circuit must return to the same electric potential.

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Potential Difference and Capacitors

A capacitor stores energy when a potential difference exists across its plates.

For a capacitor:

Q = CV

Therefore:

V = Q/C

As charge accumulates on the capacitor, the potential difference across it increases.

The stored energy is:

E = ½CV²

This is another example of voltage being closely connected to electrical energy.

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Everyday Voltage Examples

Different electrical systems operate at very different voltages.

Examples include:

• small batteries
• USB-powered electronics
• vehicle electrical systems
• household electricity
• transmission systems
• electronic circuits

The appropriate voltage depends on the purpose and design of the system.

Higher voltage can allow greater energy transfer per coulomb, but it can also create greater electrical hazards.

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Why High Voltage Is Used in Power Transmission

Electrical power is:

P = VI

For the same transmitted power, increasing voltage allows current to decrease.

Power lost as heating in transmission wires is:

P_loss = I²R

Therefore, reducing current greatly reduces transmission losses.

This is why electrical power is transmitted over long distances at high potential differences.

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Potential Difference and Electrical Safety

Potential difference can cause current to flow through the human body if a conducting path exists.

The current depends on factors including:

• voltage
• body resistance
• contact conditions
• current path
• duration

Therefore, voltage alone does not completely determine electrical danger.

Electrical safety requires preventing dangerous current from passing through the body.

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Common Mistakes

Mistake: Saying voltage is the amount of charge flowing.
Voltage is energy transferred per unit charge.

Mistake: Saying 1 V = 1 J.
The correct relationship is 1 V = 1 J/C.

Mistake: Confusing current with potential difference.
Current is charge flow rate; voltage is energy per charge.

Mistake: Using W = V/Q.
The correct rearrangement is W = QV.

Mistake: Forgetting to convert mA to A or μC to C.

Mistake: Saying a battery creates charge.
A battery transfers energy to charges and maintains a potential difference.

Mistake: Connecting a voltmeter in series.
A voltmeter is connected in parallel.

Mistake: Assuming higher voltage always means greater total energy use.
Total energy also depends on the charge transferred.

Mistake: Saying electrons move in the same direction as conventional current in metals.
Electron drift is opposite to conventional current.

Mistake: Ignoring energy conservation when analyzing voltage around a circuit.
Total potential rises and drops around a closed loop must balance.

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Key Terms

Potential difference — work done or energy transferred per unit charge between two points.

Voltage — common name for potential difference.

Volt (V) — SI unit of potential difference; 1 V = 1 J/C.

Charge (Q) — quantity of electric charge, measured in coulombs.

Coulomb (C) — SI unit of electric charge.

Work (W) — energy transferred when a force causes displacement or when charge moves through a potential difference.

Electric potential — electric potential energy per unit charge at a location.

Electric field — region in which a charge experiences an electrical force.

Current — rate of flow of electric charge.

Resistance — opposition to current flow.

Potential rise — increase in electric potential, such as across a source.

Potential drop — decrease in electric potential across a component transferring electrical energy.

Voltmeter — instrument used to measure potential difference.

Conventional current — current direction defined as the direction positive charge would move.

Electrical power — rate of electrical energy transfer.

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Key Takeaways

• Potential difference is work done or energy transferred per unit charge.
• Potential difference is commonly called voltage.
• V = W/Q.
• 1 V = 1 J/C.
• A potential difference establishes an electric field that can cause charges to move.
• Voltage and current are different quantities.
• Current measures charge flow per unit time.
• Voltage measures energy transfer per unit charge.
• Batteries maintain potential differences by transferring energy to charges.
• Components transfer electrical energy into other forms.
• W = QV can be used to calculate electrical energy transfer.
• Charge can be calculated using Q = It.
• Combining these gives W = VIt.
• Electrical power is P = VI.
• For an ohmic conductor, V = IR.
• A voltmeter measures potential difference and is connected in parallel.
• In series circuits, potential differences across components add to the supply voltage.
• Parallel branches have the same potential difference across them.
• In a uniform electric field, |ΔV| = Ed.
• Potential rises and drops around a complete circuit reflect conservation of energy.
• Understanding voltage as energy per coulomb provides a powerful way to analyze electrical circuits.

Check Your Understanding

1. Define potential difference.
2. What does a potential difference of 1 V mean?
3. Explain the difference between voltage and current.
4. Calculate the potential difference if 30 J of energy is transferred by 5 C of charge.
5. Calculate the energy transferred when 8 C moves through 12 V.
6. A device transfers 240 J of energy through a potential difference of 20 V. Calculate the charge transferred.
7. Explain how a potential difference can cause charges to move.
8. What energy transfer occurs when charge passes through a resistor?
9. A 9 V battery transfers 45 J of energy. Calculate the charge moved through the battery.
10. A current of 2 A flows for 30 s. Calculate the charge transferred.
11. A 12 V device draws 3 A for 20 s. Calculate the energy transferred.
12. A motor operates at 24 V and draws 4 A. Calculate its electrical power.
13. Explain why a voltmeter is connected in parallel.
14. Two resistors in series have potential differences of 3 V and 7 V. Calculate the supply voltage.
15. Three lamps are connected in parallel across a 12 V battery. What potential difference is across each lamp?
16. A uniform electric field has strength 4000 V/m. Calculate the potential difference across 0.050 m along the field.
17. Explain the difference between a potential rise and a potential drop.
18. Explain how voltage demonstrates conservation of energy around a complete circuit.
19. Explain why high voltage is useful for long-distance electrical power transmission.
20. A 230 V appliance draws 2.0 A for 5 minutes. Calculate the charge transferred and the electrical energy transferred.

Electric Potential Energy

What Is Electric Potential Energy?

Electric potential energy is energy stored because of the positions of electric charges relative to one another in an electric field.

It is usually represented by:

Eₚ or U

and is measured in:

joules (J)

Electric potential energy depends on:

• the amount of charge
• the signs of the charges
• the positions of the charges
• the electric potential at the location

A charge can gain or lose electric potential energy as it moves through an electric field.

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Electric Potential Energy and Electric Potential

Electric potential energy and electric potential are closely related, but they are not the same quantity.

Electric potential energy

Electric potential energy describes the energy associated with a particular charge at a particular location.

Unit:

joule (J)

Electric potential

Electric potential describes electric potential energy per unit charge at a location.

Unit:

volt (V)

Their relationship is:

Eₚ = qV

where:

• Eₚ = electric potential energy (J)
• q = charge (C)
• V = electric potential (V)

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Electric Potential Is a Property of the Location

Suppose a particular point in an electric field has:

V = 100 V

That electric potential exists whether we place a test charge there or not.

If we place:

q = +1 C

there:

Eₚ = qV

Eₚ = 1 × 100

Eₚ = 100 J

If instead:

q = +3 C

then:

Eₚ = 3 × 100

Eₚ = 300 J

The location still has the same potential, but the charge has a different potential energy.

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The Sign of the Charge Matters

Because:

Eₚ = qV

the sign of q affects the electric potential energy.

At:

V = +100 V

a charge:

q = +2 C

has:

Eₚ = (+2)(+100)

Eₚ = +200 J

But a charge:

q = −2 C

has:

Eₚ = (−2)(+100)

Eₚ = −200 J

Therefore, positive and negative charges behave differently in the same electric potential.

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Potential Difference and Changes in Potential Energy

When a charge moves between two locations:

ΔV = V_f − V_i

The corresponding change in electric potential energy is:

ΔEₚ = qΔV

or:

ΔEₚ = q(V_f − V_i)

where:

• ΔEₚ = change in electric potential energy
• q = moving charge
• V_f = final electric potential
• V_i = initial electric potential

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Interpreting the Sign of ΔEₚ

If:

ΔEₚ > 0

the charge has gained electric potential energy.

If:

ΔEₚ < 0

the charge has lost electric potential energy.

If:

ΔEₚ = 0

there has been no change in electric potential energy.

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Example: Positive Charge

A:

+2.0 C

charge moves from:

V_i = 50 V

to:

V_f = 150 V

First:

ΔV = 150 − 50

ΔV = +100 V

Then:

ΔEₚ = qΔV

ΔEₚ = (+2.0)(+100)

ΔEₚ = +200 J

The positive charge gains:

200 J

of electric potential energy.

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Example: Positive Charge Moving to Lower Potential

A:

+3.0 C

charge moves from:

200 V

to:

50 V

Potential difference:

ΔV = 50 − 200

ΔV = −150 V

Then:

ΔEₚ = (+3.0)(−150)

ΔEₚ = −450 J

The charge loses:

450 J

of electric potential energy.

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Positive Charges and Electric Fields

Electric field direction is defined as the direction of force on a positive test charge.

Electric fields point:

higher potential → lower potential

Therefore, when a positive charge moves naturally in the direction of the electric field:

electric potential decreases

and:

electric potential energy decreases

The lost electric potential energy can be converted into kinetic energy.

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Negative Charges Behave Differently

For a negative charge:

q < 0

Therefore, its potential energy behaves oppositely with electric potential.

A negative charge moving toward higher electric potential can actually have decreasing electric potential energy.

For example:

q = −2 C

moves from:

20 V → 80 V

Then:

ΔV = +60 V

ΔEₚ = (−2)(+60)

ΔEₚ = −120 J

The negative charge loses 120 J of electric potential energy even though it moved to a higher electric potential.

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Direction of Natural Motion

If released from rest and acted on only by the electric force:

Positive charge

tends to move:

high V → low V

Negative charge

tends to move:

low V → high V

In both cases, spontaneous motion caused by the electric force tends to move the charge toward:

lower electric potential energy

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Electric Potential Energy and Work

When the electric field does work on a charge:

W_field = −ΔEₚ

Therefore:

if electric potential energy decreases, the electric field does positive work.

If:

ΔEₚ = −50 J

then:

W_field = +50 J

The electric field has transferred 50 J of energy from electric potential energy into another form.

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External Work

If an external force moves a charge slowly against the electric force, the external force can increase the system's electric potential energy.

For an ideal slow movement:

W_external = ΔEₚ

For example, if moving a charge requires:

20 J

of external work against the electric force, the electric potential energy increases by:

20 J

assuming no other energy transfers.

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Gravitational Analogy

Electric potential energy can be compared with gravitational potential energy.

If you lift an object:

external work → increased gravitational potential energy

If the object falls:

gravitational potential energy → kinetic energy

Similarly, separating or rearranging electric charges can store electric potential energy.

When charges are released:

electric potential energy → kinetic energy

However, there is an important difference:

electric charges can be positive or negative

while ordinary gravitational mass is positive.

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Like Charges

Like charges repel.

Consider two positive charges.

To push them closer together, an external force must work against their repulsion.

Therefore:

separation decreases → electric potential energy increases

If released, the charges accelerate apart.

Their electric potential energy decreases while their kinetic energy increases.

---

Opposite Charges

Opposite charges attract.

Consider a positive charge and a negative charge.

To pull them apart, an external force must work against their attraction.

Therefore:

separation increases → electric potential energy increases

If released, opposite charges naturally accelerate toward each other.

Their electric potential energy decreases.

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Potential Energy of Two Point Charges

For two point charges:

Eₚ = kQq/r

where:

• Eₚ = electric potential energy (J)
• k = 8.99 × 10⁹ N·m²/C²
• Q and q = charges (C)
• r = separation between charges (m)

The signs of both charges must be included.

---

Two Positive Charges

If:

Q > 0

and:

q > 0

then:

Qq > 0

Therefore:

Eₚ > 0

Two like positive charges have positive electric potential energy relative to infinite separation.

---

Two Negative Charges

If:

Q < 0

and:

q < 0

then:

Qq > 0

Therefore:

Eₚ > 0

Two negative charges also have positive potential energy relative to infinite separation.

The important feature is that the charges have the same sign.

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Opposite Charges

If:

Q > 0

and:

q < 0

then:

Qq < 0

Therefore:

Eₚ < 0

Opposite charges have negative electric potential energy relative to infinite separation.

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Why Can Potential Energy Be Negative?

The zero of electric potential energy is often defined when charges are infinitely far apart.

For opposite charges, energy must be supplied to separate them completely.

Their bound arrangement therefore has less energy than the zero-energy reference at infinity.

That is why:

Eₚ < 0

is physically meaningful.

Negative potential energy does not mean "negative energy exists" in a problematic sense. It means the system has less energy than the chosen reference state.

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Example: Two Positive Charges

Two charges are:

Q = +2.0 μC

q = +3.0 μC

separated by:

r = 0.50 m

Use:

Eₚ = kQq/r

Eₚ = (8.99 × 10⁹)(2.0 × 10⁻⁶)(3.0 × 10⁻⁶)/0.50

Eₚ ≈ 0.108 J

Therefore:

Eₚ ≈ +0.108 J

The positive sign is expected because the charges have the same sign.

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Example: Opposite Charges

Suppose:

Q = +2.0 μC

q = −3.0 μC

r = 0.50 m

Then:

Eₚ = (8.99 × 10⁹)(2.0 × 10⁻⁶)(−3.0 × 10⁻⁶)/0.50

Eₚ ≈ −0.108 J

The negative sign is expected because the charges have opposite signs.

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How Potential Energy Changes with Distance

For two point charges:

Eₚ = kQq/r

Therefore, the magnitude of potential energy follows:

|Eₚ| ∝ 1/r

As separation increases, the potential energy approaches:

0

when the charges become extremely far apart.

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Like Charges Moving Apart

For like charges:

Eₚ is positive.

As they move farther apart:

Eₚ decreases toward zero

If they are free to move, their lost electric potential energy can become kinetic energy.

Therefore:

electric PE decreases → kinetic energy increases

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Opposite Charges Moving Together

For opposite charges:

Eₚ is negative.

As they move closer:

Eₚ becomes:

more negative

This also represents a decrease in electric potential energy.

The decrease can become kinetic energy.

---

Electric Potential Energy in a Uniform Field

Between parallel charged plates, the electric field can be approximately uniform.

For a charge moving through a potential difference:

ΔEₚ = qΔV

If the uniform field is known:

|ΔV| = Ed

so the magnitude of the potential-energy change can be related to:

|ΔEₚ| = |q|Ed

The signs and direction of motion must still be considered carefully.

---

Example Between Parallel Plates

A positive charge:

q = +4.0 × 10⁻⁶ C

moves through a potential decrease of:

250 V

Therefore:

ΔV = −250 V

Use:

ΔEₚ = qΔV

ΔEₚ = (4.0 × 10⁻⁶)(−250)

ΔEₚ = −1.0 × 10⁻³ J

The charge loses:

1.0 mJ

of electric potential energy.

---

Equipotential Lines and Potential Energy

An equipotential line connects points having the same electric potential.

If a charge moves along an equipotential:

ΔV = 0

Therefore:

ΔEₚ = qΔV

and:

ΔEₚ = 0

The charge's electric potential energy does not change.

---

Conservation of Energy

One of the most important applications of electric potential energy is the conservation of energy.

If only the electric force acts:

Eₚ,i + K_i = Eₚ,f + K_f

where:

• Eₚ = electric potential energy
• K = kinetic energy

Therefore:

ΔK = −ΔEₚ

A decrease in electric potential energy produces an equal increase in kinetic energy.

---

Charge Released from Rest

Suppose a charge begins at rest.

Then:

K_i = 0

If it loses:

50 J

of electric potential energy:

ΔEₚ = −50 J

Then:

ΔK = +50 J

Therefore:

K_f = 50 J

The charge's electric potential energy has been converted into kinetic energy.

---

Example: Finding Kinetic Energy

A:

+5.0 × 10⁻⁶ C

charge is released from rest and moves through a potential drop of:

2000 V

Potential-energy change:

ΔEₚ = qΔV

ΔEₚ = (5.0 × 10⁻⁶)(−2000)

ΔEₚ = −0.010 J

Therefore:

ΔK = +0.010 J

Since it started from rest:

K_f = 0.010 J

---

Finding Speed from Potential Difference

Kinetic energy is:

K = ½mv²

If a particle starts from rest and loses electric potential energy:

½mv² = −ΔEₚ

For an appropriate potential difference:

½mv² = |qΔV|

Therefore:

v = √(2|qΔV|/m)

This equation can be used to find the speed of charged particles accelerated through a known potential difference when non-relativistic physics is appropriate.

---

Example: Accelerating a Charged Particle

A particle has:

q = +2.0 × 10⁻⁶ C

m = 0.004 kg

and is accelerated from rest through a potential drop of:

100 V

Energy gained:

K = |qΔV|

K = (2.0 × 10⁻⁶)(100)

K = 2.0 × 10⁻⁴ J

Now:

K = ½mv²

2.0 × 10⁻⁴ = ½(0.004)v²

v² = 0.10

v ≈ 0.316 m/s

Therefore:

v ≈ 0.32 m/s

---

Electrons Accelerated Through a Voltage

Electrons are commonly accelerated using electric fields.

Although electrons are negative, the energy gained by an electron accelerated through a potential difference of magnitude V can be written:

K gained = e|ΔV|

where:

e = 1.602 × 10⁻¹⁹ C

If an electron is accelerated through:

100 V

then it gains:

100 eV

of kinetic energy.

---

The Electron Volt

An electron volt (eV) is a unit of energy.

It is defined as the energy change of one elementary charge moving through a potential difference of 1 V.

1 eV = 1.602 × 10⁻¹⁹ J

Therefore:

1 keV = 1000 eV

1 MeV = 10⁶ eV

1 GeV = 10⁹ eV

Electron volts are especially useful in atomic, nuclear, and particle physics.

---

Conservation of Energy with External Work

If an external force also does work, mechanical energy of the charge-field system does not necessarily remain constant by itself.

A useful energy relationship is:

W_external = ΔK + ΔEₚ

when other energy transfers are absent.

If a charge is moved slowly so its kinetic energy does not change:

ΔK = 0

Therefore:

W_external = ΔEₚ

---

Example: Moving Against the Field

A:

+3.0 μC

charge is slowly moved from:

100 V

to:

600 V

Potential change:

ΔV = 600 − 100

ΔV = +500 V

Potential-energy change:

ΔEₚ = qΔV

ΔEₚ = (3.0 × 10⁻⁶)(500)

ΔEₚ = +1.5 × 10⁻³ J

Therefore, an external force must supply:

1.5 mJ

of work if there are no other energy transfers.

---

Electric Potential Energy in Circuits

Electric potential energy also helps explain energy transfer in circuits.

A battery transfers energy to charges.

When those charges move through circuit components, their electrical energy is transferred into forms such as:

• thermal energy
• light
• kinetic energy
• sound

For example, a potential difference of:

12 V

means:

12 J of energy per coulomb

is transferred between those points.

---

Batteries and Energy

A battery does not create charge.

Instead, chemical reactions inside the battery transfer energy to charges and maintain a potential difference.

For example, if:

Q = 5 C

passes through a:

12 V

battery, the energy transferred is:

E = QV

E = 5 × 12

E = 60 J

The chemical energy store of the battery decreases while electrical energy is transferred through the circuit.

---

Capacitors and Electric Potential Energy

A capacitor stores energy in its electric field.

The energy stored by a capacitor is:

E = ½CV²

where:

• E = stored energy
• C = capacitance
• V = potential difference

Alternative forms include:

E = ½QV

and:

E = Q²/(2C)

The stored energy can later be transferred when the capacitor discharges.

---

Example: Capacitor Energy

A capacitor has:

C = 200 μF

and:

V = 20 V

Convert:

C = 200 × 10⁻⁶ F

Use:

E = ½CV²

E = ½(200 × 10⁻⁶)(20²)

E = 0.040 J

Therefore:

E = 0.040 J

---

Energy Transformations in Electric Fields

Electric potential energy can be transformed into many other forms.

Examples include:

Charged particle accelerating

electric potential energy → kinetic energy

Electric motor

electrical energy → kinetic energy + thermal energy

Lamp

electrical energy → light + thermal energy

Heater

electrical energy → thermal energy

Capacitor discharge

stored electric-field energy → electrical energy → other forms

Energy is transferred and transformed, but total energy remains conserved.

---

Electric Potential vs Electric Potential Energy

This distinction is essential.

---

Potential vs Potential Energy Example

Suppose points A and B have:

V_A = 100 V

V_B = 300 V

The potential difference is:

ΔV = 200 V

This potential difference is the same regardless of what test charge is considered.

However:

For q = +1 C

ΔEₚ = (+1)(200)

ΔEₚ = +200 J

For q = −1 C

ΔEₚ = (−1)(200)

ΔEₚ = −200 J

Same electric field.

Same potential difference.

Different potential-energy change because the charges have different signs.

---

Choosing the Correct Equation

Use:

Eₚ = qV

when finding potential energy at a location.

Use:

ΔEₚ = qΔV

when a charge moves between two electric potentials.

Use:

Eₚ = kQq/r

for the potential energy of two point charges.

Use:

W_field = −ΔEₚ

when calculating work done by the electric field.

Use:

Eₚ,i + K_i = Eₚ,f + K_f

when applying conservation of energy.

Use:

K = ½mv²

when connecting energy changes to particle speed.

Use:

E = ½CV²

when calculating energy stored in a capacitor.

---

Multi-Step Example

A positive particle has:

q = +4.0 μC

m = 0.002 kg

It starts from rest at:

V_i = 500 V

and moves to:

V_f = 100 V

Find the potential difference

ΔV = 100 − 500

ΔV = −400 V

Find the change in potential energy

ΔEₚ = qΔV

ΔEₚ = (4.0 × 10⁻⁶)(−400)

ΔEₚ = −1.6 × 10⁻³ J

Find the kinetic energy gained

ΔK = −ΔEₚ

ΔK = +1.6 × 10⁻³ J

Since the particle started from rest:

K_f = 1.6 × 10⁻³ J

Find the final speed

K = ½mv²

1.6 × 10⁻³ = ½(0.002)v²

v² = 1.6

v ≈ 1.26 m/s

---

Common Mistakes

Mistake: Treating electric potential and electric potential energy as the same quantity.
Potential is measured in volts; potential energy is measured in joules.

Mistake: Forgetting the charge in Eₚ = qV.
Potential energy depends on the charge.

Mistake: Ignoring the sign of a negative charge.
The sign of q is essential when using ΔEₚ = qΔV.

Mistake: Assuming higher electric potential always means higher potential energy.
This is true for positive charges but not necessarily for negative charges.

Mistake: Saying electric potential energy is always positive.
It can be positive, zero, or negative depending on the system and reference.

Mistake: Saying opposite charges have positive potential energy relative to infinity.
Their interaction potential energy is negative.

Mistake: Saying a charge gains kinetic energy without losing another form of energy.
Energy must be conserved.

Mistake: Using Eₚ = kQq/r².
Potential energy varies as 1/r, not 1/r².

Mistake: Assuming moving along an equipotential changes potential energy.
Along an equipotential, ΔV = 0 and therefore ΔEₚ = 0.

---

Key Terms

Electric potential energy — energy associated with the positions of charges in an electric field.

Electric potential — electric potential energy per unit charge at a location.

Potential difference — difference in electric potential between two locations.

Volt — unit of electric potential; 1 V = 1 J/C.

Joule — SI unit of energy.

Electric field — region in which an electric charge experiences an electrical force.

Work — energy transferred when a force causes displacement.

Conservation of energy — principle that total energy cannot be created or destroyed, only transferred or transformed.

Kinetic energy — energy associated with motion.

Point charge — idealized electric charge concentrated at one location.

Equipotential — line or surface containing points of equal electric potential.

Electron volt — unit of energy equal to 1.602 × 10⁻¹⁹ J.

Reference level — position or configuration chosen to have zero potential energy.

---

Key Takeaways

• Electric potential energy is energy associated with the positions of charges in an electric field.
• Electric potential energy is measured in joules.
• Electric potential and electric potential energy are different quantities.
• Electric potential is energy per unit charge.
• Eₚ = qV connects electric potential and potential energy.
• Changes in potential energy are calculated using ΔEₚ = qΔV.
• The sign of the charge must always be considered.
• Positive charges naturally move toward lower electric potential when acted on only by the electric field.
• Negative charges naturally move toward higher electric potential.
• Both movements correspond to decreasing electric potential energy.
• The electric field does positive work when electric potential energy decreases.
• W_field = −ΔEₚ.
• Like charges have positive interaction potential energy relative to infinite separation.
• Opposite charges have negative interaction potential energy relative to infinite separation.
• For two point charges, Eₚ = kQq/r.
• Moving along an equipotential produces no change in electric potential energy.
• If only the electric force acts, ΔK = −ΔEₚ.
• Lost electric potential energy can become kinetic energy.
• Capacitors store energy in electric fields.
• Conservation of energy provides a powerful method for solving electric-field problems.

Check Your Understanding

1. Define electric potential energy.
2. State the SI unit of electric potential energy.
3. Explain the difference between electric potential and electric potential energy.
4. A +3 C charge is at a location with electric potential 20 V. Calculate its electric potential energy.
5. A −2 C charge is at a location with electric potential +50 V. Calculate its electric potential energy.
6. Write the equation relating change in potential energy to charge and potential difference.
7. A +4 C charge moves from 20 V to 70 V. Calculate its change in electric potential energy.
8. Repeat Question 7 for a −4 C charge.
9. Explain why a positive charge naturally tends to move toward lower electric potential.
10. Explain why a negative charge can move toward higher electric potential while still losing potential energy.
11. Describe how the potential energy of two positive charges changes as they are pushed closer together.
12. Describe how the potential energy of opposite charges changes as they move closer together.
13. Write the equation for the electric potential energy of two point charges.
14. Why can electric potential energy be negative?
15. What happens to electric potential energy when a charge moves along an equipotential line?
16. A charge loses 0.50 J of electric potential energy and no other forces do work. How much kinetic energy does it gain?
17. A +5 μC charge moves through a potential drop of 600 V. Calculate its change in electric potential energy.
18. Explain how conservation of energy applies to a charged particle accelerating in an electric field.
19. A particle starts from rest and loses 2.0 × 10⁻³ J of electric potential energy. What is its final kinetic energy?
20. Explain how batteries and capacitors demonstrate the storage and transfer of electrical energy.

Capacitors and Energy Storage

What Is a Capacitor?

A capacitor is an electrical component designed to store separated electric charge and energy in an electric field.

A simple capacitor consists of:

• two conducting plates
• an insulating material or gap between the plates
• connections allowing the plates to be connected to a circuit

The insulating region prevents charge from flowing directly from one plate to the other.

---

Structure of a Parallel-Plate Capacitor

The simplest model is the parallel-plate capacitor.

It consists of two conducting plates separated by a distance:

d

When connected to a voltage source:

• electrons are transferred onto one plate
• electrons are removed from the other plate
• one plate becomes negatively charged
• the other becomes positively charged

If one plate has:

+Q

the other has:

−Q

The capacitor as a whole can remain electrically neutral because the charges are separated rather than created.

---

The Electric Field Between the Plates

The separated charges create an electric field between the plates.

For an ideal parallel-plate capacitor, away from the edges, this field is approximately:

• uniform
• perpendicular to the plates
• directed from the positive plate toward the negative plate

Energy is stored in this electric field.

---

What Does a Capacitor Store?

It is common to say that a capacitor "stores charge."

More precisely, a capacitor stores:

separated charge and energy in its electric field

The two plates carry equal amounts of opposite charge:

+Q and −Q

The important feature is the separation of charge.

---

Capacitance

Capacitance measures how much charge a capacitor can store for each volt of potential difference.

It is represented by:

C

and defined by:

C = Q/V

where:

• C = capacitance (F)
• Q = magnitude of charge on either plate (C)
• V = potential difference across the capacitor (V)

Rearranging:

Q = CV

and:

V = Q/C

---

The Farad

The SI unit of capacitance is the:

farad (F)

From:

C = Q/V

we obtain:

1 F = 1 C/V

A capacitor with capacitance:

1 F

stores:

1 C

of charge on each plate when the potential difference is:

1 V

One farad is quite large for many ordinary electronic circuits.

---

Common Capacitance Units

Smaller units are commonly used:

1 mF = 10⁻³ F

1 μF = 10⁻⁶ F

1 nF = 10⁻⁹ F

1 pF = 10⁻¹² F

For example:

470 μF = 470 × 10⁻⁶ F

= 4.70 × 10⁻⁴ F

Unit conversion is very important in capacitor calculations.

---

Example: Finding Capacitance

A capacitor stores:

Q = 0.003 C

when connected across:

V = 12 V

Use:

C = Q/V

C = 0.003/12

C = 0.00025 F

Therefore:

C = 250 μF

---

Example: Finding Charge

A:

200 μF

capacitor is connected across:

12 V.

Convert:

C = 200 × 10⁻⁶ F

Use:

Q = CV

Q = (200 × 10⁻⁶)(12)

Q = 2.4 × 10⁻³ C

Therefore:

Q = 2.4 mC

---

What Determines Capacitance?

For a parallel-plate capacitor:

C = εA/d

where:

• C = capacitance
• ε = permittivity of the material between the plates
• A = overlapping plate area
• d = plate separation

This equation shows that capacitance depends on the capacitor's physical construction.

---

Increasing Plate Area

From:

C = εA/d

we see:

C ∝ A

Increasing plate area increases capacitance.

Larger plates can hold more separated charge for the same potential difference.

Therefore:

larger area → larger capacitance

---

Changing Plate Separation

Capacitance is inversely proportional to plate separation:

C ∝ 1/d

Therefore:

smaller separation → larger capacitance

If the plate separation is doubled:

capacitance is halved

provided everything else remains constant.

---

Dielectrics

The insulating material between capacitor plates is called a:

dielectric

Possible dielectric materials include:

• air
• ceramic
• plastic
• paper
• mica
• oxide layers

A dielectric increases capacitance compared with a vacuum for the same geometry.

---

Why a Dielectric Increases Capacitance

The molecules or atoms in a dielectric become polarized by the capacitor's electric field.

This polarization partially opposes the field produced by the free charges on the plates.

As a result, the capacitor can store more free charge for a given potential difference.

Therefore:

dielectric → greater capacitance

---

Relative Permittivity

The capacitance may also be written:

C = ε₀εᵣA/d

where:

• ε₀ = permittivity of free space
• εᵣ = relative permittivity of the dielectric

A material with a larger εᵣ generally produces a larger capacitance for the same plate geometry.

---

Charging a Capacitor

A capacitor can be charged by connecting it to a voltage source through a conducting path.

Initially, an uncharged capacitor has:

Q = 0

and:

V_C = 0

When connected to a battery, charge begins moving through the circuit.

---

What Happens During Charging?

As the capacitor charges:

• one plate gains electrons
• the other loses electrons
• Q increases
• V_C increases
• the electric field becomes stronger
• stored energy increases
• charging current gradually decreases

Eventually, in an ideal DC circuit:

V_C ≈ V_supply

and:

I ≈ 0

The capacitor is then considered fully charged.

---

Why Does the Current Decrease?

At the beginning:

V_C = 0

so the battery produces the greatest potential difference across the resistor.

The current is therefore largest.

As the capacitor charges:

V_C increases

This leaves less voltage across the resistor.

Since:

I = V_R/R

the current decreases.

Eventually:

V_C ≈ V_supply

V_R ≈ 0

and:

I ≈ 0

---

Capacitor Voltage During Charging

The voltage across a charging capacitor rises exponentially.

The exact relationship is:

V_C = V₀(1 − e^(−t/RC))

where:

• V_C = capacitor voltage at time t
• V₀ = supply voltage
• R = resistance
• C = capacitance
• t = time

The voltage approaches the supply voltage but, mathematically, never reaches it exactly.

---

Charge During Charging

Since:

Q = CV

the charge follows the same general pattern:

Q = Q_max(1 − e^(−t/RC))

where:

Q_max = CV₀

Charge rises rapidly at first and then more slowly.

---

Current During Charging

Charging current decreases exponentially:

I = I₀e^(−t/RC)

where:

I₀ = V₀/R

The current is greatest immediately after the circuit is connected.

It then decreases toward zero.

---

The RC Time Constant

The rate at which a capacitor charges or discharges depends on:

R

and:

C

The time constant is:

τ = RC

where:

• τ = time constant (s)
• R = resistance (Ω)
• C = capacitance (F)

Because:

Ω × F = s

the unit of RC is seconds.

---

Meaning of One Time Constant

After one time constant during charging:

V_C ≈ 63% of its final value

and:

Q ≈ 63% of its final value

The current has fallen to approximately:

37% of its initial value

After about:

5τ

the capacitor is usually considered essentially fully charged.

---

Example: Time Constant

A circuit contains:

R = 20 kΩ

C = 100 μF

Convert:

R = 20,000 Ω

C = 100 × 10⁻⁶ F

Then:

τ = RC

τ = (20,000)(100 × 10⁻⁶)

τ = 2.0 s

After approximately:

2 s

the capacitor has reached about 63% of its final voltage.

After approximately:

10 s

or 5τ, it is essentially fully charged.

---

Effect of Resistance

If capacitance remains constant:

larger R → larger τ

Therefore, increasing resistance causes the capacitor to:

• charge more slowly
• discharge more slowly

Resistance limits the current flowing into or out of the capacitor.

---

Effect of Capacitance

If resistance remains constant:

larger C → larger τ

A larger-capacitance capacitor takes longer to charge or discharge through the same resistance.

It can also store more charge at the same voltage because:

Q = CV

---

Discharging a Capacitor

A charged capacitor can release its stored energy by connecting its plates through a conducting path.

During discharge:

• charge decreases
• capacitor voltage decreases
• electric field weakens
• stored energy decreases
• current decreases in magnitude

---

Discharge Equation

For an ideal RC discharge:

V_C = V₀e^(−t/RC)

Similarly:

Q = Q₀e^(−t/RC)

The magnitude of the current also decreases exponentially.

After one time constant:

V_C ≈ 37% of its initial value

and:

Q ≈ 37% of its initial value

---

Charging vs Discharging

Both processes are exponential rather than linear.

---

Energy Stored in a Capacitor

A capacitor stores energy in its electric field.

The stored energy is:

E = ½CV²

where:

• E = energy stored (J)
• C = capacitance (F)
• V = potential difference (V)

Alternative forms are:

E = ½QV

and:

E = Q²/(2C)

These equations are equivalent because:

Q = CV

---

Why Is There a Factor of ½?

While a capacitor is charging, its voltage does not remain constant.

It begins at:

0 V

and rises to:

V

For an ideal capacitor, the average voltage during the charging process is:

V/2

Therefore:

energy = charge × average voltage

E = Q(V/2)

so:

E = ½QV

Since:

Q = CV

we obtain:

E = ½CV²

---

Example: Energy Stored

A capacitor has:

C = 500 μF

and is charged to:

V = 12 V

Convert:

C = 500 × 10⁻⁶ F

Use:

E = ½CV²

E = ½(500 × 10⁻⁶)(12²)

E = 0.036 J

Therefore:

E = 0.036 J

---

Voltage Has a Strong Effect on Stored Energy

Since:

E = ½CV²

energy depends on the square of voltage.

If voltage doubles:

V → 2V

then:

E → ½C(2V)²

E → 4(½CV²)

Therefore:

doubling voltage quadruples stored energy

for a capacitor with constant capacitance.

---

Example: Doubling Voltage

Suppose a capacitor stores:

5 J

at:

100 V

If the voltage increases to:

200 V

then:

E_new/E_old = (200/100)²

= 4

Therefore:

E_new = 20 J

The voltage doubled, but the energy quadrupled.

---

Energy During Discharge

When a capacitor discharges, its stored electric-field energy is transferred into other forms.

If discharged through a resistor:

electric-field energy → thermal energy

If discharged through a lamp:

electric-field energy → light + thermal energy

If discharged through a motor:

electric-field energy → kinetic + thermal + sound energy

Energy is conserved.

---

Energy Does Not Decrease Linearly

Because:

E = ½CV²

and voltage decreases exponentially during discharge, stored energy also decreases nonlinearly.

If capacitor voltage falls to half its original value:

V_new = V/2

then:

E_new = ½C(V/2)²

Therefore:

E_new = E_original/4

Half the voltage means only one-quarter of the original stored energy remains.

---

Capacitors in Parallel

Capacitors connected in parallel have the same voltage across them.

Their total capacitance is:

C_total = C₁ + C₂ + C₃ + ...

For example:

C₁ = 10 μF

C₂ = 20 μF

C₃ = 30 μF

Then:

C_total = 10 + 20 + 30

C_total = 60 μF

Adding capacitors in parallel increases total capacitance.

---

Capacitors in Series

For capacitors in series:

1/C_total = 1/C₁ + 1/C₂ + 1/C₃ + ...

For two capacitors:

C_total = C₁C₂/(C₁ + C₂)

The total capacitance of capacitors in series is less than the smallest individual capacitance.

---

Example: Capacitors in Series

Suppose:

C₁ = 6 μF

C₂ = 3 μF

Then:

C_total = (6 × 3)/(6 + 3)

C_total = 18/9

C_total = 2 μF

---

Capacitors vs Batteries

Capacitors and batteries can both store energy, but they operate differently.

Capacitors are particularly useful when energy must be stored or released quickly.

---

Camera Flash

A camera flash is a classic capacitor application.

The capacitor:

1. charges gradually from the battery
2. stores electrical energy
3. releases the energy very rapidly
4. produces a short, intense flash of light

The battery alone may not be able to deliver the required power as quickly.

---

Power-Supply Smoothing

Capacitors are widely used in electronic power supplies.

After AC has been rectified, the voltage may fluctuate.

A capacitor can:

• charge when voltage rises
• discharge when voltage falls

This helps produce a smoother output voltage.

This process is called:

smoothing or filtering

---

Timing Circuits

The predictable charging and discharging of capacitors makes them useful for timing.

Because:

τ = RC

designers can choose R and C to control how quickly voltage changes.

Applications include:

• flashing lights
• delay circuits
• timers
• oscillators
• control circuits

---

Capacitors in Electronic Devices

Capacitors are found throughout modern electronics.

Applications include:

• computers
• smartphones
• televisions
• audio equipment
• power supplies
• communication systems
• vehicles
• sensors
• motor circuits
• medical equipment

A single electronic device may contain many different types of capacitors.

---

Decoupling Capacitors

Digital circuits can produce rapid changes in current.

Small capacitors placed close to integrated circuits can provide charge during these brief changes.

They also help reduce unwanted voltage fluctuations and electrical noise.

These are often called:

decoupling capacitors

or:

bypass capacitors

They are extremely common on computer circuit boards.

---

Capacitors in Audio Systems

Capacitors can help separate signals of different frequencies.

They may be used in:

• filters
• amplifiers
• speaker crossover networks
• tone-control circuits

Their response depends on factors including:

• capacitance
• signal frequency
• surrounding resistance

---

Capacitors in Motors

Some AC motors use capacitors to help create appropriate phase relationships between currents.

Applications include:

• fans
• pumps
• compressors
• air-conditioning systems

These may use:

• start capacitors
• run capacitors

depending on the motor design.

---

Supercapacitors

A supercapacitor is designed to provide much larger capacitance than many conventional capacitors.

Supercapacitors can:

• charge quickly
• discharge quickly
• survive many cycles
• provide large bursts of power

Applications can include:

• regenerative braking
• backup power
• energy recovery
• industrial systems
• short-term energy storage

They generally store less energy per unit mass than many batteries but can deliver power much more rapidly.

---

Regenerative Braking

Some transportation systems recover kinetic energy during braking.

Instead of converting all kinetic energy into thermal energy at the brakes:

kinetic energy → electrical energy → stored energy

Capacitors or supercapacitors can sometimes store this recovered energy rapidly.

The energy can later help accelerate the vehicle again.

---

Capacitor Voltage Ratings

Real capacitors have a maximum safe operating voltage called the:

voltage rating

For example, a capacitor might be marked:

470 μF, 25 V

This means:

• capacitance = 470 μF
• maximum intended operating voltage is approximately 25 V according to its specification

Operating above the voltage rating can damage the dielectric and may cause capacitor failure.

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Polarized Capacitors

Some capacitors, especially many electrolytic capacitors, are polarized.

They have designated:

• positive terminal
• negative terminal

Connecting a polarized capacitor incorrectly can damage it and may create a safety hazard.

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Capacitor Safety

A capacitor can remain charged even after a circuit has been switched off.

Therefore:

disconnected does not necessarily mean discharged

Large or high-voltage capacitors can store dangerous amounts of energy.

They should be handled using appropriate electrical safety procedures.

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High-Voltage Energy Example

Suppose:

C = 1000 μF

V = 400 V

Convert:

C = 0.001 F

Stored energy:

E = ½CV²

E = ½(0.001)(400²)

E = 80 J

Therefore:

E = 80 J

This demonstrates why large high-voltage capacitors can remain hazardous even after the power supply is disconnected.

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Analyzing a Capacitor Problem

A useful approach is:

Identify what is known

For example:

C = 200 μF

V = 10 V

R = 50 kΩ

Convert units

C = 200 × 10⁻⁶ F

R = 50,000 Ω

Choose the required equation

For charge:

Q = CV

For energy:

E = ½CV²

For charging time:

τ = RC

Calculate carefully

Then check that the answer has a reasonable unit and magnitude.

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Multi-Step Example

A capacitor has:

C = 400 μF

It is connected to:

V = 20 V

through:

R = 10 kΩ

Maximum charge

Q = CV

Q = (400 × 10⁻⁶)(20)

Q = 8.0 × 10⁻³ C

or:

Q = 8.0 mC

Stored energy

E = ½CV²

E = ½(400 × 10⁻⁶)(20²)

E = 0.080 J

Time constant

τ = RC

τ = (10,000)(400 × 10⁻⁶)

τ = 4.0 s

After approximately:

4 s

the capacitor has reached about 63% of its final voltage and charge.

After approximately:

20 s

it is essentially fully charged.

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Choosing the Correct Equation

Use:

C = Q/V

when relating capacitance, charge and voltage.

Use:

Q = CV

when calculating stored charge.

Use:

C = εA/d

for an ideal parallel-plate capacitor.

Use:

E = ½CV²

when calculating stored energy from capacitance and voltage.

Use:

E = ½QV

when charge and voltage are known.

Use:

E = Q²/(2C)

when charge and capacitance are known.

Use:

τ = RC

when analyzing charging or discharging time.

Use:

V_C = V₀(1 − e^(−t/RC))

for charging voltage.

Use:

V_C = V₀e^(−t/RC)

for discharging voltage.

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Common Mistakes

Mistake: Saying a capacitor creates charge.
It separates existing charge between its plates.

Mistake: Saying energy is stored only "on the plates."
The stored energy is associated with the capacitor's electric field.

Mistake: Confusing capacitance with charge.
Capacitance measures the ability to store charge per volt.

Mistake: Using C = QV.
The correct relationship is C = Q/V.

Mistake: Forgetting to convert μF into F.
1 μF = 10⁻⁶ F.

Mistake: Assuming capacitor voltage changes instantly during ordinary RC charging.
Voltage changes progressively according to the charging curve.

Mistake: Assuming charging current remains constant.
It decreases as the capacitor charges.

Mistake: Saying a fully charged ideal capacitor continues carrying steady DC current.
In the ideal steady state, current approaches zero.

Mistake: Assuming 1τ means fully charged.
After 1τ, the capacitor is only about 63% charged.

Mistake: Assuming half the voltage means half the stored energy.
Because E ∝ V², half the voltage means one-quarter of the energy.

Mistake: Assuming a disconnected capacitor is automatically safe.
It may remain charged.

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Key Terms

Capacitor — component that stores separated electric charge and energy in an electric field.

Capacitance — amount of charge stored per unit potential difference.

Farad (F) — SI unit of capacitance; 1 F = 1 C/V.

Plate — conducting surface of a capacitor on which charge accumulates.

Dielectric — insulating material between capacitor plates.

Permittivity — property describing how a material responds to an electric field.

Charging — process of increasing separated charge and stored energy in a capacitor.

Discharging — process of releasing stored electrical energy from a capacitor.

Time constant — characteristic charging or discharging time given by τ = RC.

RC circuit — circuit containing resistance and capacitance.

Stored energy — energy associated with the electric field of a charged capacitor.

Voltage rating — maximum intended operating potential difference across a capacitor.

Polarized capacitor — capacitor designed to be connected with a particular polarity.

Supercapacitor — very-high-capacitance device capable of rapid charge and discharge.

Decoupling capacitor — capacitor used to reduce voltage fluctuations and electrical noise near electronic components.

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Key Takeaways

• A capacitor consists essentially of two conductors separated by an insulator.
• Capacitors store separated charge and energy in an electric field.
• The plates carry equal and opposite charges.
• Capacitance is defined by C = Q/V.
• The unit of capacitance is the farad.
• Greater plate area increases capacitance.
• Smaller plate separation increases capacitance.
• A dielectric can increase capacitance.
• For parallel plates, C = εA/d.
• During charging, charge and capacitor voltage increase while current decreases.
• During discharging, charge and voltage decrease.
• Capacitor charging and discharging are exponential processes.
• The RC time constant is τ = RC.
• After 1τ, a charging capacitor reaches about 63% of its final voltage.
• After 1τ of discharge, about 37% of the original voltage remains.
• Capacitor energy can be calculated using E = ½CV².
• Stored energy depends on the square of voltage.
• Capacitors in parallel add directly.
• Capacitors in series combine reciprocally.
• Capacitors are used for energy storage, filtering, timing, smoothing, signal processing and rapid power delivery.
• Real capacitors have voltage, polarity and safety limitations.

Check Your Understanding

1. Describe the basic structure of a capacitor.
2. Explain what happens to charge when a capacitor is connected to a battery.
3. Where is the energy of a charged capacitor stored?
4. Define capacitance.
5. What does a capacitance of 1 F mean?
6. A capacitor stores 0.006 C at 12 V. Calculate its capacitance.
7. Calculate the charge stored by a 300 μF capacitor connected across 20 V.
8. State how increasing plate area affects capacitance.
9. State how increasing plate separation affects capacitance.
10. Explain why inserting a dielectric increases capacitance.
11. Describe how capacitor voltage changes during charging.
12. Describe how current changes during charging.
13. Define the RC time constant.
14. Calculate τ for R = 25 kΩ and C = 200 μF.
15. Approximately what percentage of its final voltage has a capacitor reached after one time constant?
16. Approximately what percentage of its original voltage remains after one time constant of discharge?
17. Calculate the energy stored by a 500 μF capacitor charged to 20 V.
18. If the voltage across a capacitor doubles, by what factor does its stored energy change?
19. Explain how a capacitor can be used in a camera flash or electronic power supply.
20. Explain why a high-voltage capacitor may remain dangerous even after the power supply has been disconnected.